Modular Construction of Free Hyperplane Arrangements

Abstract

In this article, we study the freeness of hyperplane arrangements. One of the most investigated arrangements is a graphic arrangement. Stanley proved that a graphic arrangement is free if and only if the corresponding graph is chordal, and Dirac showed that a graph is chordal if and only if the graph is obtained by ''gluing'' complete graphs. We will generalize Dirac's construction to simple matroids with modular joins introduced by Ziegler and show that every arrangement whose associated matroid is constructed in the manner mentioned above is divisionally free. Moreover, we apply the result to arrangements associated with gain graphs and arrangements over finite fields.